Understanding Engineering Mathematics

Now, provided− 1 <r<1wehavern→0asn→∞,so lim n→∞ Sn= a 1 −r Thus, the sum to infinity of the geometric series is S= a 1 −r ...

Answer 3 2 [ 1 − ( 1 3 )n] ,^32 14.10 Tests for convergence As noted in the previous section, if we can calculate the general fo ...

Problem 14.12 Apply the comparison test to the series (i) U= 1 Y 1 2! Y 1 3! Y··· ( =e^1 − 1 ) (ii) U= 1 2 Y 1 4 Y 1 6 Y··· (i) ...

D’Alembert’s ratio test Lest, as an engineer, you doubt the relevance of all this pure mathematics to your studies, a few words ...

So the series converges, as we already know from Problem 14.12. It is in fact the series fore^1 −1. (ii) For 1+ 2 1 + 22 2! + 23 ...

(ii) Apply the ratio test: if l=lim n→∞ ∣ ∣ ∣ ∣ un+ 1 un ∣ ∣ ∣ ∣ then the series converges ifl<1 and diverges ifl>1. (iii) ...

A Maclaurin’s series is thus just a Taylor series about the origin. Note that by changing the variable,X=x−awe can always conver ...

Problem 14.15 Obtain the Maclaurins series for the functions (i)ex (ii) cosx (i)exis the easy one – it just keeps repeating on d ...

So the ratio test tells us that this series converges for all finitex. It is, of course, the series forex. (ii) Forx− x^2 2 + x^ ...

14.12 Reinforcement (i) Find the values of the function( 3 x+ 1 )/xwhenxhas the values 10, 100, 1000, 1,000,000. (ii) What limi ...

9.Write down the first four terms of the sequence whosenth term is: (i) n (ii) (n^2 + 3 n)/ 5 n (iii) cos^12 nπ (iv) ( 1 +(− 1 ) ...

(i) 1+ 1 2 + 1 22 + 1 23 + 1 24 +··· (ii) 1− 1 3 + 1 32 − 1 33 +··· (iii) 1+ 3 + 5 + 7 +··· (iv) 1+ 1 + 1 2! + 1 3! + 1 4! +··· ...

21.Expand( 1 + 2 x) 3 (^2) as far as the term inx^2. How many terms of the series would be required to give( 1. 02 ) 3 (^2) corr ...

Fory=f(x)=x^2 obtain a linear approximation tof(x), and hence evaluate( 1. 01 )^2 approximately. 3.You may have heard talk of ‘c ...

(vii)a,ar,ar^2 ,ar^3 , (divergent if|r|>1, limit=aifr=1, convergent to 0 if |r|<1 and oscillating ifr=− 1 ) (viii)− x^3 3 ...

(i) 1 (D) (ii) (− 1 )n+^1 n (D) (iii) n(. 9 )n−^1 (C) (iv) (− 1 )n+^1 √ n (C) (v) n^3 (. 95 )n−^1 (C) (vi) 2 n− 1 2 n (D) (vii ...

15 Ordinary Differential Equations Ordinary differential equations bring together all the calculus that we have done so far in t ...

Objectives In this chapter you will find: definitions and terminology for differential equations initial and boundary condition ...

Dividing byk: dn kdt = dn d(kt) =n and if we now introduce new variables: x=kt,y=n it becomes dy dx =y( 15. 2 ) This is a good e ...

The solution obtained is, in this case, themost general solution–there are no others. To fixAwe would have to specify an extra ...